SCIENTIFIC BACKGROUND

High Reynolds number flow is a classical research theme
that retains its vitality at several levels, from
real-world applications, through physical and
computational modeling, up to rigorous mathematical
analysis. There are two reasons for the continued
relevance of this topic. The first is the ubiquity of
such flows in situations of practical interest, such as
blood flow in large caliber vessels, fluid-structure
interaction, aerodynamics, geophysical and astrophysical
flow modeling. The second issue is that, despite of half
a century of vigorous efforts, there is still a lack of
systematic understanding how different scales interact
to form the inertial range from a smooth initial
condition. The description of the behavior of solutions
of the Navier-Stokes equations at high Reynolds number
is at the heart of the problem, and surprisingly,
mathematical analysis seems to be a promising route for
gaining insight. Is singularity formation of
incompressible flows at high Reynolds number necessary
for the formation of the inertial range in a turbulent
flow? or is the dynamical generation of extremely small
but finite scales sufficient for this purpose? The choice
of the singularity problem for the incompressible Navier-Stokes equation as one of the seven Millennium
prize problems highlights the fundamental role that
mathematical analysis may yet play in this subject,
while attesting to the quality of the mathematical
challenge posed by problems in this area.

This is the second CSCAMM workshop on this topic,
following our Spring 2004 meeting.

SCIENTIFIC CONTENT

This field has seen substantial progress in several
independent directions. Let us cite a few prominent
examples: the understanding of the interplay between the
local geometric properties of the vorticity field and
vortex stretching, the use of the Kato method applied to
the Navier-Stokes equations in identifying critical
spaces for well-posedness, the solution of the water
wave problem and related research on interfacial
dynamics, the mathematical understanding of the problem
of boundary layers. A wide variety of methods have been
employed, from classical functional analysis and
operator theory to modern harmonic analysis and
geometric measure theory.

Several interesting problems remain open. Beyond the
singularity problem we highlight the uniqueness of weak
solutions with p-th power integrable vorticity,
existence of weak solutions with vortex sheet initial
data for the two dimensional ideal flow equations and
the convergence of the vanishing viscosity
approximation in the presence of boundaries. This
workshop is intended as a forum where the recent
progress is examined from the point of view of
understanding the large time behavior of the
incompressible flows at high Reynolds number. The
participants will include a representative sample of
researchers active in the field of mathematical analysis
of incompressible flows, together with a few specialists
in fluid dynamics.